Observability inequalities and measurable sets

Abstract

This paper presents two observability inequalities for the heat equation over Ω x (0, T). In the first one, the observation is from a subset of positive measure in Ω x (0; T), while in the second, the observation is from a subset of positive surface measure on ∂Ω x (0, T). It also proves the Lebeau-Robbiano spectral inequality when Ω is a bounded Lipschitz and locally star-shaped domain. Some applications for the above-mentioned observability inequalities are provided.